Question

Michael is sitting on a Ferris wheel. He is exactly 21 feet from the center and is at the 3 o'clock position when the Ferris wheel begins moving. The bottom of the Ferris wheel is 8 feet above the ground. The Ferris wheel broke down within the first revolution of the ride when Michael was at the point (-6.454, -19.984).a. What is the measure of the angle (in radians) that has a vertex (0,0)and rays through the point (21,0) and (-6.454, -19.984)?θ= radians   b. How many feet did Michael travel along the arc before stopping?_____ feet 

Answer 1

We will solve as follows:

a. We will find the measure of the angle using the following expression:

$c^2=a^2+b^2-2ab\cos (C)$

Here a, b & c are the sides delimited by the triangle formed inside the circle and the points given and C is the angle between them. Now, we solve for C, then replace and solve[This can be seen in the following graph]:

$\Rightarrow\frac{a^2+b^2-c^2}{2ab}=\cos (C)\Rightarrow C=\cos ^{-1}(\frac{a^2+b^2-c^2}{2ab})$

Now, we determine the values of a. b and c using the points given and the vertex:

$a=\sqrt[]{(0-0)^2+(21-0)^2}^{}\Rightarrow a=21$$b=\sqrt[]{(-6.454-0)^2+(-19.984-0)^2}\Rightarrow b=21$$c=\sqrt[]{(21+6.454)^2+(0+19.984)^2}\Rightarrow c=\sqrt[]{1153.082372}\Rightarrow c\approx33.9570666$

Now, we replace the values on the formula and solve for C:

$C=\cos ^{-1}(\frac{21^2+21^2-(33.9570666)^2}{2(21)(21)})\Rightarrow C\approx107.8995792$

And that in radians is:

$C\approx\frac{107.8995791}{180}\cdot\pi\Rightarrow C\approx0.5994421061\pi\Rightarrow C\approx1.8832029185517$

So, the angle is approximately 1.89 radians.

b.

We will find the distance he traveled along the arc as follows:

$\theta=\frac{s}{r}\Rightarrow s=\theta r$

Here s is the arc length, theta the angle and r the radius, now we replace:

$\Rightarrow s=(1.89)(21)\Rightarrow s\approx39.69$

So, he traveled approximately 39.69 feet.